4.4 Georeferencing (Geometric Correction)

The purpose of georeferencing is to transform the image coordinate system (u,v), which may be distorted due to the factors discussed above, to a specific map projection (x,y) as shown in Figure 4.8. The imaging process involves the transformation of a real 3-D scene geometry to a 2-D image

Figure 4.8. Georeferencing is a transformation between the image space to the geographical coordinate space

Terms such as geometric rectification or image rectification, image-to-image registration, image-to-map registration have the following meanings:

1) Geometric rectification and image rectification recovers the imaging geometry

2) Image-to-image registration refers to transforming one image coordinate system into another image coordinating system

3) Image-to-map registration refers to transformation of one image coordinate system to a map coordinate system resulted from a particular map projection.

Georeferencing generally covers 1) and 3). It requires a transformation T:

Forward Transformation  is composed of the following transformations:

 

In order to achieve:

Every step involved in the imaging process has to be known, i.e., we need to know the inverse process of geometric transformation.

This is a complex and time consuming process. However, there is a simpler and widely-used alternative: polynomial approximation.

Coefficients a's and b's are determined by using Ground Control Points (GCPs).

For example, we can use very low order polynomials such as the affine transformation

u = ax + by + c

v = dx + ey + f

A minimum of 3 GCPs will enable us to determine the coefficients in the above equations.

In this way, we don't need to use the transformation matrix T. However, in order to make our coefficients representative of the whole image that is transformed, we have to make sure that our GCPs are well distributed all over the image.

The third choice is that we can combine the T-1 method with the polynomial technique in order to reduce the transformation errors involved in the direct transformation of T-1 (Figure 4.9).

Figure 4.9. Larger magnitude of errors may be introduced
if direct transformation is used.

This may be achieved through the following four steps:

(1) To refine imaging geometry parameters.

In T-1, due to the inaccuracies of satellite or plane positioning, polynomials are used to correct them. For platform position the following formula can be used:



We can use GCPs to refine the coefficients. Global Positioning System (GPS) and/or Inertial Navigation Systems (INS) techniques can also be used. The integration of GPS and INS with remote sensing sensors are being investigated (Schwarz, et al, 1993).

(2) Divide output grid into blocks (Figure 4.10):

Figure 4.10. In the x-y space

(3) Map the grid points using

(4) Use a low order polynomial inside each block for detailed mapping (Figure 4.11)

Figure 4.11. Further transformation from u-v space to Dx-Dy space using lower order polynomials

The choices are:

(i) Affine

(ii) Bilinear  

Why (ii) is called bilinear? This is because each coordinate can be a multiplication of two linear function of x and y.

u = (a + bx) (c + dy)


linear linear

Bilinear

Since there are four known and four unknown, therefore we can solve (i) using least squares (ii) using regular solution of an equation group. We will only show how to obtain ao, a1, a2, a3 in (ii).

For point

Similarly, we can obtain bo, b1, b2, b3.

Why do we use bilinear instead of affine? It is because the bilinear transformation guarantees the continuity from block to block in the detailed mapping. The geometric interpolation of bilinear transformation is illustrated in Figure 4.12.

Figure 4.12. Linear and bilinear transformation

Method for Determining the coefficients of a polynomial geometric transformation

We can use the least square solution for bilinear polynomials.

This is done with more than 4, say n, GCPs,

(u1, v1) (x1, y1)

(u2, v2) (x2, y2)

.

.

.

(un, vn) (xn, yn)

by substituting the n GCPs coordinates into (1) and (2) we will obtain two groups of over-determined equations

The least squares solution in matrix form:

For x

by multiplying on both sides, we have

A is the solution. Similarly we can solve bo, b1, b2, b3.

This can be applied to affine transformation and higher order polynomial transformation.